To ensure global convergence (i.e., to ensure convergence to a solution from any initial point), suitable modifications of the Newton method are needed. An example of a globally convergent variant is the so-called Levenberg-Marquardt method [1]}, [2]}. This involves a modification of Newton's search direction at each step of the method. Without this modification, however, quadratic convergence can only be ensured when the initial guess belongs to a quadratic convergence region, namely a region from which every starting point generates a quadratically convergent Newton sequence.
